Clean sphere separation with a sphere created from a cube
As Martin Z says, you don't say enough about your problem, also, what problem did you meet with the knife? To cut with the knife, press K for activate, then Z to cut through, then C to cut orthogonally

But if we keep your request, it looks like if you select an edge loop (alt right click), then scale it to 0 on the Z axis with S Z 0, then W > LoopTools > Circle, it works fine. Don't forget to activate the LoopTools addon in the Preferences.

And to separate a mesh in two parts, select the edge loop and press V.